Citations and references In developing the ideas described in this book I have looked at many thousands of books, papers and websites—and have interacted with hundreds of people (see page xiii ).

continuous form, and shows no trace of the underlying discreteness of the system; the randomness has in a sense successfully washed out essentially all the microscopic details of the system. The pictures at the top of the facing page show what happens if one uses several different underlying rules for the motion of each particle. … The top left shows three individual examples of random walks, in which each particle randomly moves one position to the left or right.

But experiments—the most direct of which are based on looking for quantization in the measured decay times of very short-lived particles—have only demonstrated continuity on scales longer than about 10 -26 seconds, and there is nothing to say that on shorter scales time is not in fact discrete.

Frequency of behavior [in multiway systems] Among multiway systems with randomly chosen rules, one finds about equal numbers that grow rapidly and die out completely.

in the top picture below. … Each of the networks illustrated at the top of the facing page consists at the lowest level of a collection of identical nodes. … The picture shows all inequivalent cases ignoring labels, but excludes networks in which there are nodes which cannot be reached by connections from other nodes.

And in fact up to 26D (with the exception of 11 through 13) all the densest packings known so far are lattices that work like this. … In all examples found so far the densest packings can always be repetitive; most can also be highly symmetrical—though in high dimensions random lattices often do not yield much worse results.

The digits that lie directly below and to the left of the original 1 at the top of the pattern correspond to the whole number part of each successive number (e.g. 3 in 3.375), while the digits that lie to the right correspond to the fractional part (e.g. 0.375 in 3.375). … These sizes are plotted at the top of the next page . … But division by 2 just does the opposite of multiplication by 2, so in base 2 it simply shifts all digits one position to the right.

On the top left the basic forms are given in so-called sequency order. On the top right they are reordered roughly so as to go systematically from coarser to finer. … The images shown here are all rescaled so that smallest values are white and largest black.

So the question is then which equivalence results need to be included in the axiom system in order that all other equivalence results can be deduced just from these. … But in some cases it turns out to be possible to establish that a particular set of axioms can successfully generate all equivalence results for a given operator—and indeed the picture at the top of the facing page shows examples of this for each of the four operators in the picture below. … (The first argument to each operator is shown on the left; the second on top.)

But as soon as one perturbs such initial conditions, one normally seems to get only complicated and seemingly random behavior, as in the top row of pictures in the second image. … The top row shows the effect of inserting a single extra black cell into various backgrounds. The bottom row shows all localized structures involving up to 25 cells supported by rule 30 on repetitive backgrounds with blocks of up to 25 cells.